Question

$\begin{vmatrix}a&b&c&d\\\ -a&b&c&d\\\ -a&-b&c&d\\\ -a&-b&-c&d\end{vmatrix} =$

• A. $8abcd$
• B. $abcd$
• C. $4abcd$
• D. $6abcd$

Answer 1

Answer: (A)

$8abcd$

Answer 2

$\begin{vmatrix}a&b&c&d\\\ -a&b&c&d\\\ -a&-b&c&d\\\ -a&-b&-c&d\end{vmatrix}$
Taking common $a, b, c$ and $d$ from $C_1, C_2, C_3$ and $C_4$ respectively, and applying $R_2 \to R_2 + R_1, R_3 \to R_1 + R_3$ and $R_4 \to R_4 + R_1$
$abcd\begin{vmatrix}1&1&1&1\\\ 0&2&2&2\\\ 0&0&2&2\\\ 0&0&0&2\end{vmatrix}$
Now expanding along $C_1$, we get
=abcd\left\\{1\begin{vmatrix}2&2&2\\\ 0&2&2\\\ 0&0&2\end{vmatrix}-0+0-0\right\\}
=abcd 1\left\\{2\left(4-0\right)-0+0\right\\}=8abcd